Just a week after agreeing on a framework for a nuclear deal, Iran’s supreme leader and the Obama administration clashed over its core elements, rekindling doubts about whether Washington and Tehran can finalize an accord by a June 30 deadline.

The supreme leader, Ayatollah Ali Khamenei, in his first public comments on the diplomacy, said on Thursday the U.S. and its negotiating partners must lift all sanctions on his country immediately upon a final deal being signed. The Obama administration has been demanding a phased repeal...